A graph class \(\mathcal {G}\)  is hereditary if \(G\in \mathcal {G}\)  implies that every induced subgraph of \(G\)  belongs to \(\mathcal {G}\) . Obviously, \(\mathcal {G}\)  is hereditary if and only if \(\mathcal {G}\)  is the class of \(\mathcal {H}\) -free graphs for some \(\mathcal {H}\) . A hereditary graph class \(\mathcal {G}\)  is \(\chi \) -bounded if there is a function \(f:\mathbb {N}\rightarrow \mathbb {N}\)  (with \(f(1)=1\)  and \(f(x)\ge x\) , for all \(x\in \mathbb {N}\) ) such that \(\chi (G)\le f(\omega (G))\)  for every \(G\in \mathcal {G}\) . The function \(f\)  is called a \(\chi \) -binding function for \(\mathcal {G}\) . Gyárfás [1]} conjectured that, if \(H\)  is any fixed tree, then the class of \(H\) -free graphs is \(\chi \) -bounded, and proved that every \(P_t\) -free graph \(G\)  satisfies \(\chi (G)\le (t-1)^{\omega (G)-1}\) . Since this \(\chi \) -binding function is exponential in \(\omega \) , researchers tried to find a linear \(\chi \) -binding function for the class of \(P_t\) -free graphs. Unfortunately, the class of \(P_t\) -free graphs has no linear \(\chi \) -binding function when \(t\ge 5\)  [2]}. However, some linear \(\chi \) -binding functions for subclasses of \(P_t\) -free graphs have been found. For example, every \(P_4\) -free graph \(G\)  satisfies \(\chi (G)=\omega (G)\)  [3]}. If a graph \(G\)  is (\(P_6, C_4\) )-free or (\(P_5, P_4\vee K_1\) )-free or (\(P_4+K_1, P_4\vee K_1\) )-free, then \(G\)  satisfies \(\chi (G)\le \lceil \frac{5\omega (G)}{4}\rceil \) ; see [4]}, [5]}, [6]}. Every (\(P_6\) , diamond)-free graph \(G\)  satisfies \(\chi (G)\le \max \lbrace 6, \omega (G)\rbrace \)  [7]}.
